Enter the height and the angle of launch into the calculator to determine the minimum initial velocity to reach the height.

Minimum Initial Velocity Formula

The following equation is used to calculate the Minimum Initial Velocity.

Vmin = SQRT ( 2*G*H / Sin^2(a) )
  • Where Vmin is the minimum initial velocity to hit height H (m/s)
  • G is the acceleration due to gravity (9.81 m/s^2)
  • H is the height to be reached (m)
  • a is the angle of the launch (degrees)

To calculate the minimum initial velocity, take the product of 2 times G times the height, divide by the sin squared of the angle, then take the square root of the result.

What is a Minimum Initial Velocity?

Definition:

A minimum initial velocity is the magnitude of a velocity needed in order for a projectile to reach a given height.

How to Calculate Minimum Initial Velocity?

Example Problem:

The following example outlines the steps and information needed to calculate Minimum Initial Velocity.

First, determine the height to be reached. In this example, the height to be reached is found to be 50m.

Next, determine the angle of the launch. For this problem, the angle of the launch is found to be 30 degrees.

Finally, calculate the Minimum Initial Velocity using the formula above:

Vmin = SQRT ( 2*G*H / Sin^2(a) )

Vmin = SQRT ( 2*9.81*50 / Sin^2(30deg) )

Vmin = 62.64 m/s

FAQ

What factors affect the minimum initial velocity required for a projectile?
The minimum initial velocity required for a projectile to reach a certain height is primarily affected by the height of the target (H), the angle of launch (a), and the acceleration due to gravity (G). The angle of launch significantly influences the trajectory and the distance a projectile can cover, while gravity affects how quickly it returns to the ground.

Can the minimum initial velocity be different for the same height but different angles?
Yes, the minimum initial velocity can vary for the same height but different angles of launch. This is because the angle of launch affects the vertical and horizontal components of the initial velocity. A steeper angle requires more vertical velocity to reach the same height, while a shallower angle spreads the velocity more evenly between vertical and horizontal components.

How does air resistance affect the calculation of minimum initial velocity?
The formula provided for calculating minimum initial velocity assumes a vacuum, meaning it does not take air resistance into account. In real-world scenarios, air resistance opposes the motion of the projectile, requiring a higher initial velocity to reach the same height compared to calculations without air resistance. The effect of air resistance becomes more significant at higher velocities and with projectiles that have larger surface areas.